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Sum-product number

A sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it algebraically, given an integer n that is l digits long in base b (with dx representing the xth digit), if

then n is a sum-product number in base b. In base 10, the only sum-product numbers are 0, 1, 135, 144 (sequence A038369 in the OEIS). Thus, for example, 144 is a sum-product number because 1 + 4 + 4 = 9, and 1 × 4 × 4 = 16, and 9 × 16 = 144.

1 is a sum-product number in any base, because of the multiplicative identity. 0 is also a sum-product number in any base, but no other integer with significant zeroes in the given base can be a sum-product number. 0 and 1 are also unique in being the only single-digit sum-product numbers in any given base; for any other single-digit number, the sum of the digits times the product of the digits works out to the number itself squared.

Any integer shown to be a sum-product number in a given base must, by definition, also be a Harshad number in that base.

In binary, 0 and 1 are the only sum-product numbers. The following table lists the sum-product numbers in bases up to 40 (using A−Z to represent digits 10 to 35):

The finiteness of the list for base 10 was proven by David Wilson. First he proved that a base 10 sum-product number will not have more than 84 digits. Next, he ruled out numbers with significant zeroes. Thereafter he concentrated on digit products of the forms 2i3j7k{\displaystyle 2^{i}3^{j}7^{k}} or 3i5j7k{\displaystyle 3^{i}5^{j}7^{k}}, which the previous constraints reduce to a set small enough to be testable by brute force in a reasonable period of time.

From Wilson's proof, Raymond Puzio developed the proof that in any positional base system there is only a finite set of sum-product numbers. First he observed that any number n of length l must satisfy n≥bl−1{\displaystyle n\geq b^{l-1}}. Second, since the largest digit in the base represents b - 1, the maximum possible value of the sum of digits of n is l(b−1){\displaystyle l(b-1)} and the maximum possible value of the product of digits is (b−1)l{\displaystyle (b-1)^{l}}. Multiplying the maximum possible sum by the maximum possible product gives l(b−1)l+1{\displaystyle l(b-1)^{l+1}}, which is an upper bound of the value of any sum-product number of length l. This suggests that l(b−1)l+1≥n≥bl−1{\displaystyle l(b-1)^{l+1}\geq n\geq b^{l-1}}, or dividing both sides, l(b−1)2≥(b/(b−1))l−1{\displaystyle l(b-1)^{2}\geq (b/(b-1))^{l-1}}. Puzio then deduced that, because of the growth of exponential function, this inequality can only be true for values of l less than some limit, and thus that there can only be finitely many sum-product numbers n.

In Roman numerals, the only sum-product numbers are 1, 2, 3, and possibly 4 (if written IIII).